Mathematics Colloquia and Seminars

Brauer groups and rationality of quotient varieties

Colloquium

Speaker:

Tihomir Petrov, UC Irvine

Location:

693 Kerr

Start time:

Mon, Nov 1 2004, 4:10PM

We will discuss some questions related to the stable rationality of
quotient varieties $V/G$ where $G$ is an algebraic group and $V$
is a faithful complex linear representation of $G$. The first examples of
nonrational and even nonstably rational varieties $V/G$ were obtained by
showing that a birational invariant, the so-called unramified Brauer
group, is nontrivial for some series of groups. In fact,
this invariant coincides with the cohomological (or Grothendieck's) Brauer
group of a smooth projective model for $V/G$. However, the unramified
point of view enables one to dispense with the construction of an explicit
smooth model, and even with the existence of such a model. This obstruction
is the first of the series of birational invariants of $V/G$ constructed
via group cohomology. Some recent results when $G$ is a finite simple group
will be presented.